Chapter 11.4, Problem 58E

a)

To determine

To evaluate: the model for the average cost per unit produced.

Answer to Problem 58E

Model for the average cost per unit produced is given by $\frac{\text{Cost function}}{\text{Number of binoculars produced}}$ .

Explanation of Solution

Given:

The cost functions for a new supermarket to recycle x tons of organic material is given as $C=50x+2700$ where C is the cost (in dollars).

Cost function is given as,

  $C=50x+2700$

Average cost is given by,

  $\begin{array}{c}\text{Average cost = }\frac{\text{Cost function}}{\text{Number of tons of organic material}}\\ =\frac{50x+2700}{x}\end{array}$

So,

Model for the average cost per ton of organic material recycled is given by $\frac{\text{Cost function}}{\text{Number of tons of organic material}}$ .

b)

To determine

To evaluate: the average costs per unit when x = 100 tons and x =1000 tons .

Answer to Problem 58E

The average costs per unit when x = 1000 is 52 dollars and the average costs per unit when x = 5000 is 188 dollars .

Explanation of Solution

Given:

The cost functions for a new supermarket to recycle x tons of organic material is given as $C=50x+2700$ where C is the cost (in dollars).

Cost function is given as,

  $C=50x+2700$

Average cost is given by,

  $\begin{array}{c}\text{Average cost = }\frac{\text{Cost function}}{\text{Number of tons of organic material}}\\ =\frac{50x+2700}{x}\end{array}$

So,

Average costs per unit when x = 100 tons,

  $\begin{array}{c}\text{Average cost = }\frac{50x+2700}{x}\\ =\frac{50\left(100\right)+2700}{100}\\ =77\text{ dollars}\end{array}$

Average costs per unit when x = 1000 tons ,

  $\begin{array}{c}\text{Average cost = }\frac{50x+2700}{x}\\ =\frac{50\left(1000\right)+2700}{1000}\\ =52.7\text{ dollars}\end{array}$

So,

The average costs per unit when x = 100 is 77 dollars and the average costs per unit when x = 1000 is 52.7 dollars .

c)

To determine

To evaluate: the limit of the average cost function as x approaches infinity.

Answer to Problem 58E

The limit of average cost function is 50.

Explanation of Solution

Given:

The cost functions for a new supermarket to recycle x tons of organic material is given as $C=50x+2700$ where C is the cost (in dollars).

Cost function is given as,

  $C=50x+2700$

Average cost is given by,

  $\begin{array}{c}\text{Average cost = }\frac{\text{Cost function}}{\text{Number of tons of organic material}}\\ =\frac{50x+2700}{x}\end{array}$

So,

Finding the limit,

  $\begin{array}{c}\underset{x\to \infty }{\lim }\frac{50x+2700}{x}=\underset{x\to \infty }{\lim }50+\frac{2700}{x}\\ =50\end{array}$

The meaning of the limit in this context is when recycling x tons of organic material is infinity, then the cost per ton becomes constant that is 50.

So, the limit of average cost function is 50.